Theorem sseqtrrd 3240

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2213	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3239	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  sseqtrrid  3252  fnfvima  5842  tfrlemiubacc  6439  tfr1onlemubacc  6455  tfrcllemubacc  6468  rdgivallem  6490  nnnninf  7254  nninfwlpoimlemg  7303  ccatass  11102  swrdval2  11142  dfphi2  12657  ctinf  12916  imasaddfnlemg  13261  imasaddvallemg  13262  subsubm  13430  subsubg  13648  subsubrng  14091  subsubrg  14122  lidlss  14353  toponss  14613  ssntr  14709  iscnp3  14790  cnprcl2k  14793  tgcn  14795  tgcnp  14796  ssidcn  14797  cncnp  14817  txcnp  14858  imasnopn  14886  hmeontr  14900  blssec  15025  blssopn  15072  xmettx  15097  metcnp  15099  plyaddlem1  15334  plymullem1  15335  plycoeid3  15344  nnsf  16144  nninfsellemsuc  16151